Local Behavior of Solutions to Subelliptic Problems with Hardy Potential on Carnot Groups

  • Annunziata Loiudice


We determine the exact behavior at the singularity of solutions to semilinear subelliptic problems of the type \(-\Delta _{\mathbb {G}}u -\mu \dfrac{\psi ^2}{d^2} u =f(\xi ,u)\) in \(\Omega \), \(u=0\) on \(\partial \Omega \), where \(\Delta _{\mathbb {G}}\) is a sub-Laplacian on a Carnot group \(\mathbb {G}\) of homogeneous dimension Q, \(\Omega \) is an open subset of \(\mathbb {G}\), \(0\in \Omega \), d is the gauge norm on \(\mathbb {G}\), \(\psi :=|\nabla _{\mathbb {G}}d|\), where \(\nabla _{\mathbb {G}}\) is the horizontal gradient associated with \(\Delta _{\mathbb {G}}\), f has at most critical growth and \(0\le \mu < \overline{\mu }\), where \(\overline{\mu }=\left( \frac{Q-2}{2} \right) ^2\) is the best Hardy constant on \(\mathbb {G}\).


Subelliptic critical problem Hardy potential asymptotic behavior Carnot groups 

Mathematics Subject Classification

35J70 35J75 35B40 


  1. 1.
    Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie groups and potential theory for their Sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin (2007)zbMATHGoogle Scholar
  2. 2.
    Bonfiglioli, A., Uguzzoni, F.: Nonlinear Liouville theorems for some critical problems on H-type groups. J. Funct. Anal. 207, 161–215 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brezis, H., Kato, T.: Remarks on the Schrödinger operator with singular complex potential. J. Math. Pures et Appl. 58, 137–151 (1979)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cao, D., Han, P.: Solutions to critical elliptic equations with multi-singular inverse square potentials. J. Differ. Equ. 224, 332–372 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, J.: Exact local behavior of positive solutions for a semilinear elliptic equation with Hardy term. Proc. Am. Math. Soc. 132, 3225–3229 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, J.: On a semilinear elliptic equation with singular term and Hardy-Sobolev critical growth. Math. Nachr. 280(8), 838–850 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cowling, M., Dooley, A.H., Korányi, A., Ricci, F.: \(H\)-type groups and Iwasawa decompositions. Adv. Math. 87, 1–41 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D’Ambrosio, L.: Hardy-type inequalities related to degenerate elliptic differential operators. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 4(5), 451–486 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    D’Ambrosio, L., Mitidieri, E.: Quasilinear elliptic equations with critical potentials. Adv. Nonlinear Anal. 6(2), 147–164 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Danielli, D., Garofalo, N., Phuc, N.C.: Hardy-Sobolev type inequalities with sharp constants in Carnot-Carathéodory spaces. Potential Anal. 34, 223–242 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dou, J., Han, Y., Zhang, S.: A class of Caffarelli-Kohn-Nirenberg inequalities on the \(H\)-type groups. Rend. Sem. Mat. Univ. Padova 132, 249–266 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Felli, V., Ferrero, A., Terracini, S.: Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential. J. Eur. Math. Soc. 13, 119–174 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Felli, V., Schneider, M.: Compactness and existence results for degenerate critical elliptic equations. Commun. Contemp. Math. 7, 37–73 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Felli, V., Terracini, S.: Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity. Commun. Partial Differ. Equ. 31, 469–495 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ferrero, A., Gazzola, F.: Existence of solutions for singular critical growth semilinear elliptic equations. J. Differ. Equ. 177, 494–522 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Folland, G.B.: Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13, 161–207 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Folland, G.B., Stein, E.: Hardy spaces on homogeneous groups, Mathematical Notes, 28, University Press, Princeton, N.J. (1982)Google Scholar
  18. 18.
    Garofalo, N., Lanconelli, E.: Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Ann. Inst. Fourier (Grenoble) 40(2), 313–356 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Garofalo, N., Lanconelli, E.: Existence and nonexistence results for semilinear equations on the Heisenberg group. Indiana Univ. Math. J. 41, 71–98 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Garofalo, N., Vassilev, D.: Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot Groups. Math. Ann. 318, 453–516 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Han, P.: Asymptotic behavior of solutions to semilinear elliptic equations with Hardy potential. Proc. Am. Math Soc. 135, 365–372 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jannelli, E.: The role played by space dimension in elliptic critical problems. J. Differ. Equ. 156, 407–426 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jannelli, E., Loiudice, A.: Critical polyharmonic problems with singular nonlinearities. Nonlinear Anal. 110, 77–96 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lanconelli, E., Uguzzoni, F.: Asymptotic behavior and non existence theorems for semilinear Dirichlet problems involving critical exponent on unbounded domains of the Heisenberg group, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1, 139-168 (1998)Google Scholar
  25. 25.
    Lanconelli, E., Uguzzoni, F.: Non-existence results for semilinear Kohn-Laplace equations in unbounded domains. Commun. Partial Differ. Equ. 25, 1703–1739 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Loiudice, A.: Semilinear subelliptic problems with critical growth on Carnot groups. Manuscripta Math. 124, 247–259 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Loiudice, A.: Asymptotic behaviour of solutions for a class of degenerate elliptic critical problems. Nonlinear Anal. 70(8), 2986–2991 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Loiudice, A.: Critical growth problems with singular nonlinearities on Carnot groups. Nonlinear Anal. 126, 415–436 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mokrani, H.: Semilinear subelliptic equations on the Heisenberg group with a singular potential. Commun. Pure Appl. Math. 8, 1619–1636 (2009)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Smets, D.: Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities. Trans. Am. Math. Soc. 375, 2909–2938 (2005)CrossRefzbMATHGoogle Scholar
  31. 31.
    Terracini, S.: On positive entire solutions to a class of equations with singular coefficient and critical exponent. Adv. Differ. Equ. 1, 241–264 (1996)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di BariBariItaly

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